# Crystal structures

Entry by Emily Redston, AP 225, Fall 2011

Crystalline materials are characterized by repeating (or periodic) arrays of atoms over large atomic distances. Typically we think about these materials in the context of solids, though there are also liquid crystals. Crystallization is the process of formation of crystal structures, where the atoms will position themselves in a repetitive three-dimensional structure upon solidification. The system is said to display long-range order. If a solid does not have long-range order, it is considered amorphous. The crystal structure of a material is the way in which atoms, ions, or molecules are spatially arranged. There is an extremely large number of difference crystal structures that all have long-range order. Some of these are relatively simple structure (e.g. metals) while others can be exceedingly complex (e.g. ceramic and polymeric materials).

The term lattice is oftentimes used to describe a three-dimensional array of points coinciding with atom positions.

crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry. Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. The symmetry properties of the crystal are embodied in its space group.

The unit cell is the smallest structure that repeats itself by translation through the crystal. We construct these symmetrical units with the hard spheres.

The atomic order in crystallline order in crystalline solids indicates that small groups of atoms form a repetitive pattern. Thus, in describing crystal structures, it is often convenient to sub-divide the structure in small repeat entities called unit cells. A unit cell is choisen to represent the symmetry of the crystal structure, wherein all the atom positions in the crystal may be generated by translation of the unit cell integral distances along each of its edges. Thus, the unit cell is the basis structural unit of the crystal structure and defines the crystal the crystal structure by virtue of its geometry and the atom positions within.

Briefly, since they are relevant to hard sphere colloidal packing, I will go over two essential structures: face-centered cubic and hexagonal close packed. For hard-sphere models, you typically end up with relatively large numbers of nearest neighbors and dense atomic packings due to the minimal restrictions as to the number and position of nearest-neighbor atoms.

The face-centered cubic system (F) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell (1⁄8 × 8 from the corners plus 1⁄2 × 6 from the faces).

The crystal structure of a material or the arrangement of atoms within a given type of crystal structure can be described in terms of its unit cell. The unit cell is a small box containing one or more atoms, a spatial arrangement of atoms. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. The crystal structure has a three-dimensional shape. The unit cell is given by its lattice parameters, which are the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (xi , yi , zi) measured from a lattice point.

Many crystal structures are based on a close-packing of atoms, or of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one another in energy, and it may be difficult to predict which form will be preferred from first principles.

Figure 1 – The hcp lattice (left) and the fcc lattice (right). The outline of each respective Bravais lattice is shown in red. The letters indicate which layers are the same. There are two "A" layers in the hcp matrix, where all the spheres are in the same position. All three layers in the fcc stack are different. Note the fcc stacking may be converted to the hcp stacking by translation of the upper-most sphere, as shown by the dashed outline. (http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres)

There are two simple regular lattices that achieve this highest average density. They are called face-centered cubic (fcc) (also called cubic close packed) and hexagonal close-packed (hcp), based on their symmetry. Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; they differ in how the sheets are stacked upon one another.

In both the fcc and hcp arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres (octahedral) and two smaller gaps surrounded by four spheres (tetrahedral). The distances to the centers of these gaps from the centers of the surrounding spheres is \scriptstyle \sqrt{\frac{3}{2}} for the tetrahedral, and \scriptstyle \sqrt2 for the octahedral, when the sphere radius is 1.

Relative to a reference layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius.

The most regular ones are:

   fcc = ABCABCA (every third layer is the same)
hcp = ABABABA (every other layer is the same)


In close-packing, the center-to-center spacing of spheres in the x–y plane is a simple honeycomb-like tessellation with a pitch (distance between sphere centers) of one sphere diameter. The distance between sphere centers, projected on the z (vertical) axis, is:

   \text{pitch}_Z = \sqrt{6} \cdot {d\over 3}\approx0.81649658 d,


where d is the diameter of a sphere; this follows from the tetrahedral arrangement of close-packed spheres.

The coordination number of hcp and fcc is 12 and its atomic packing factor (APF) is the number mentioned above, 0.74.

## References

[1] Callister, William D. Materials Science and Engineering: an Introduction. New York: John Wiley & Sons, 2007.